02141493v53n1p73

Publ. Mat. 53 (2009), 73–82
ON FIXED POINTS OF AUTOMORPHISMS OF
NON-ORIENTABLE UNBORDERED KLEIN SURFACES
Grzegorz Gromadzki
Abstract
In 1973, Macbeath found a general formula for the number of
points fixed by an arbitrary orientation preserving automorphism
of a Riemann surface X. It was given in terms of a group G of conformal automorphisms of X and the ramification data of the covering X → X/G, which corresponds to the so called universal covering transformation group. In these terms, for the case of a cyclic
group of automorphisms of an unbordered non-orientable Klein
surface, the formula was given later by Izquierdo and Singerman
and here we find formulas valid for an arbitrary (finite) group G
of automorphisms.
1. Introduction
It is not difficult to see, that the set of fixed points of an automorphism of a compact Riemann surface consists either of isolated points
or simple closed Jordan curves called ovals (and sometimes mirrors).
The last case occurs only for anticonformal involutions, in the literature
known as symmetries. In 1973, Macbeath found in [14] a formula for
the number of points fixed by an arbitrary automorphism of a Riemann
surface X in terms of the group of conformal automorphisms of X and
the ramification data of its action, which corresponds to the so called
universal covering transformation group. Later we found (in [7] —see
also [8]) a similar formula for the number of ovals of a symmetry.
It is worth mentioning here that the possible number of ovals of a symmetry of a Riemann surface is given by the classical Harnack-Weichold
Theorem [10], [17]. A useful role is also played by the method of HoareSingerman [11], concerning non-normal subgroups of NEC-groups, which
2000 Mathematics Subject Classification. Primary: 30F; Secondary: 14H.
Key words. Automorphisms of Riemann and Klein surfaces, fixed-point set, Fuchsian
and NEC-groups, uniformization.
Supported by SAB2005-0049 and by MTM2005-01637 of the Spanish Ministry of
Sciences and Education.
74
G. Gromadzki
allowed the relations between the total number of ovals of two symmetries and the order of their product to be found in [4], and the nature
of the set of fixed points of involutions of compact Klein surfaces with
boundary in [3].
The case of non-orientable surfaces is essentially different, since both
types of fixed points may occur simultaneously. For a cyclic group G
of automorphisms, Izquierdo and Singerman found [12] formulas for the
number of isolated fixed points and the number of ovals of any automorphisms from G, in terms of the universal covering transformation group.
Here we find such formulas for an arbitrary group of automorphisms of
such a surface. We also give some illustrative examples.
2. Preliminaries
We shall use combinatorial methods based on the Riemann uniformization theorem and theory of Fuchsian and NEC-groups as in [5],
where the reader can find necessary concepts and facts together with
the precise references to the original sources. By an unbordered nonorientable Klein surface we mean a non-orientable compact topological
surface with a dianalytic structure which, roughly speaking, differs from
the classical analytic one by the fact, that the complex conjugation is
allowed for transition functions of charts, see [1] for preciseness. The
principal role in combinatorial study of such surfaces is nowadays being
played by the counterpart of the Riemann uniformization theorem, by
which such a surface X can be represented as the orbit space H/Γ of the
hyperbolic plane, with respect to the action of some, so called surface
NEC-group [1]. The notion of a Klein surface has already been known to
Klein himself, but until seventies such surfaces have been studied either
as algebraic curves with real equations or as Riemann surfaces together
with a single antiholomorphic involution by considering automorphisms
commuting with it. By [1], for a surface given as such orbit space, its
group of automorphisms G can be represented as the factor group Λ/Γ
for some other NEC-group Λ and the pair (H, Λ) is called universal covering transformation group for the action (G, X).
An NEC-group is a discrete subgroup of the group of isometries G
of the hyperbolic plane H, including those reversing orientation with a
compact orbit space. If Λ contains no orientation preserving elements of
finite order, then it is called surface NEC-group. Using fundamental regions, Macbeath and Wilkie [13], [18] associated to every NEC-group Λ
the so called signature, which determines its algebraic structure. It has
Fixed Points of Automorphisms of Klein Surfaces
75
the form
(1)
(g; ±; [m1 , . . . , mr ]; {(n11 , . . . , n1s1 ), . . . , (nk1 , . . . , nksk )})
and mentioned above surface groups have signatures (g; ±; [−]; {−}).
The numbers mi ≥ 2 are called the proper periods, the brackets
(ni1 , . . . , nisi ), the period cycles, the numbers nij ≥ 2 are the link periods and g ≥ 0 is said to be the orbit genus of Λ. The orbit space H/Λ
is a surface with k boundary components, orientable or not according
to the sign being + or −, having topological genus g and the canonical
projection H → H/Λ is a covering ramified over r interior points with
ramification indices mi and over si points lying on each boundary component with ramification indices nij . A group with the signature (1) has
a presentation given by generators:
(a) xi ,
(b) cij ,
i = 1, . . . , r,
i = 1, . . . , k, j = 0, . . . , si ,
(elliptic elements)
(hyperbolic reflections)
(c) ei ,
i = 1, . . . , k,
(boundary generators)
(d) ai , bi , i = 1, . . . , g if the sign is +, (hyperbolic translations)
di ,
i = 1, . . . , g if the sign is −, (glide reflections)
and relations
i
= 1,
(1) xm
i
(2) cisi =
i = 1, . . . , r,
e−1
i ci0 ei ,
i = 1, . . . , k,
(3) c2ij = 1,
(cij−1 cij )nij = 1,
i = 1, . . . , k, j = 0, . . . , si ,
i = 1, . . . , k, j = 1, . . . , si ,
−1
−1 −1
(4) x1 . . . xr e1 . . . ek a1 b1 a−1
1 b1 . . . ag bg ag bg = 1 or
x1 . . . xr e1 . . . ek d2i . . . d2g = 1.
Any system of generators of an NEC-group, which satisfies the above
relations, will be called a canonical system of generators and it is known,
that every element of finite order in Λ is conjugate either to a canonical
reflection or to a power of some canonical elliptic element xi or else to a
power of the product of two consecutive canonical reflections cij−1 , cij .
For every NEC-group we have an associated fundamental region,
whose hyperbolic area µ(Λ) depends only on the group and for a group
with signature (1) it is given by
!
si
k X
r
X
X
(1 − 1/nij ) ,
(1 − 1/mi ) + 1/2
(2) 2π εg + k − 2 +
i=1
i=1 i=1
76
G. Gromadzki
where ε = 2 or 1 according to the sign being + or −. It is known that an
abstract group with the presentation given by the above generators and
relations can be realized as an NEC-group with the signature (1) if and
only if (2) is positive. Finally, if Λ′ is a subgroup of finite index in an
NEC-group Λ, then it is an NEC-group itself and the Hurwitz-Riemann
formula (known also as the Hurwitz formula) says that
(3)
[Λ : Λ′ ] = µ(Λ′ )/µ(Λ).
3. On fixed points of an automorphism of unbordered
non-orientable Klein surfaces
Let X = H/Γ be a non-orientable, unbordered Klein surface, where
Γ is an NEC-group with signature (g; −; [−]; {−}) and let the action
of G on X be defined by an epimorphism θ : Λ → G, where Λ is an
NEC-group with signature (1). We start this principal section of the
paper with the lemma describing the nature of the set of points fixed by
an automorphism of X.
Lemma 3.1. The set of points fixed by an automorphism of an unbordered, non-orientable Klein surface consists of isolated points and ovals.
Proof: The canonical projection π : H → X is a non-ramified covering
and the action of G on X is defined by
gx = π(λh) if g = θ(λ), x = π(h).
So if ϕ = θ(λ), then x = π(h) is its fixed point if and only if λh = γh for
some γ ∈ Γ, which means that γ −1 λ fixes h. But then γ −1 λ is either an
elliptic element or a reflection, say c, since these are the only hyperbolic
isometries with fixed points. In the first case x is obviously an isolated
fixed point. Now, if in the second case ℓ is the axis of c, then on the
one hand π(ℓ) is the fixed-point set of ϕ, while on the other hand it is
homeomorphic to a circle, as π is an unramified covering.
Theorem 3.2. The number of isolated fixed points of ϕ ∈ G on X is
given by the formula
X
X
|NG (hϕi)|
1/mi +
1/nij ,
where N stands for the normalizer and the sums are taken respectively
over canonical elliptic generators and consecutive canonical reflections
for which ϕ is conjugate to a power of θ(xi ) and θ(cij−1 cij ) respectively.
Fixed Points of Automorphisms of Klein Surfaces
77
Theorem 3.3. An involution σ of X has
X
[C(G, θ(c)) : θ(C(Λ, c))]
ovals, where C denotes the centralizer and the sum is taken over nonconjugate canonical reflections of Λ, whose images under θ are conjugate
to σ in G.
The proof of Theorem 3.2: If ϕ has order n, then the number of its fixed
points equals the number of the conjugacy classes of cyclic subgroups of
order n of Γϕ = θ−1 (hϕi). Clearly each such subgroup is generated by
an elliptic element. However, each elliptic element is conjugate either to
a power of some canonical elliptic generator or to a power of the product
of two consecutive canonical reflections of Λ and we say that fixed points
of ϕ are produced by xi or cij−1 cij in each of these cases respectively. We
have to find how many fixed points of ϕ is produced by each element xi
and by each product cij−1 cij .
Assume first, that xi produces fixed points of ϕ. Then, since conjugate
elements have the same number of fixed points, by exchanging ϕ with a
suitable conjugation, we may assume that xni i ∈ Γϕ for ni = mi /n. Now
wxni i w−1 ∈ Γϕ if and only if w ∈ θ−1 (NG (hϕi)). Observe however, that
wxni i w−1 and w′ xni i w′−1 are conjugate in Γϕ if and only if w−1 γw′ ∈
NΛ (hxni i i) = hxi i for some γ ∈ Γϕ , which means that w−1 w′ ∈ hxi iΓϕ .
Thus xi produces
[θ−1 (NG (hϕi)) : hxi iΓϕ ] = [θ−1 (NG (hϕi))/Γ : hxi iΓϕ /Γ] = |NG (hϕi)|/mi
fixed points of ϕ.
Similarly, assume that cij−1 cij produces fixed points of ϕ. Then
again we may assume that (cij−1 cij )mij ∈ Γϕ for mij = nij /n. Also
w(cij−1 cij )mij w−1 ∈ Γϕ if and only if w ∈ θ−1 (NG (hϕi)). On the other
hand w(cij−1 cij )mij w−1 and w′ (cij−1 cij )mij w′−1 are conjugate in Γϕ
if and only if w−1 γw′ belongs to NΛ (h(cij−1 cij )mij i) = hcij−1 cij i, for
some γ ∈ Γϕ , which means that w−1 w′ ∈ hcij−1 cij iΓϕ . Hence cij−1 cij
produces
[θ−1 (NG (hϕi)) : hcij−1 cij iΓϕ ] = [θ(θ−1 (NG (hϕi))) : θ(hcij−1 cij iΓϕ )]
= [NG (hϕi) : hθ(cij−1 cij )i]
= |NG (hϕi)|/nij
fixed points of ϕ.
78
G. Gromadzki
The proof of Theorem 3.3: The proof here is similar to the proof of the
formula for the number of ovals of a symmetry of Riemann surface
from [7]. We have to count reflections of Λ, which are in Γσ = θ−1 (hσi)
but are not conjugate there. If Fix(σ) contains an oval, then σ is conjugate to θ(ci ) for some canonical reflection ci of Λ. Now, without loss of
generality, we may assume that θ(ci ) = σ, since conjugate symmetries
have the same number of ovals. Observe that for w ∈ Λ, cw
i ∈ Γσ if and
only if w ∈ θ−1 (C(G, θ(ci ))). Denote the last by Ci and observe that it
normalizes Γσ . Thus for v, w ∈ Ci , the reflections cvi and cw
i of Γσ are
conjugate in Γσ if and only if w−1 v ∈ C(Λ, ci )Γσ . As a consequence,
conjugates of ci give rise to
[Ci : C(Λ, ci )Γσ ] = [C(G, θ(ci )) : θ(C(Λ, ci ))]
empty period cycles in Γσ .
−1
′
Let now cw
=
i′ ∈ Γσ for some i 6= i and w ∈ Λ. Then θ(w)θ(ci′ )θ(w)
σ and so
(4)
wCi′ w−1 = Ci .
Indeed, if λ ∈ wCi′ w−1 then θ(w)−1 θ(λ)θ(w) ∈ C(G, θ(ci′ )). Thus θ(λ)
centralizes σ and so λ ∈ Ci . Conversely, if λ ∈ Ci , then θ(λ) normalizes σ. Hence θ(λ)θ(w)θ(ci′ )θ(w)−1 θ(λ)−1 = σ, which in turn means
that θ(w)−1 θ(λ)θ(w) centralizes θ(ci′ ) and so λ ∈ wCi′ w−1 as claimed.
Furthermore, cvi′ ∈ Γσ if and only if vw−1 ∈ Ci . Indeed, if this is the
case, then θ(v)θ(ci′ )θ(v)−1 = σ and so θ(w)−1 θ(v) ∈ C(Λ, θ(ci′ )). Thus
w−1 v ∈ Ci′ , which gives vw−1 ∈ wCi′ w−1 = Ci . The converse is similar
and we omit it.
′
Finally, given u, u′ ∈ Ci and v = uw, v ′ = u′ w, the reflections cvi′ , cvi′
are conjugate in Γσ if and only if v −1 v ′ ∈ C(Λ, ci′ )w−1 Γσ w = C(Λ, ci′ )Γ,
which means that u−1 u′ ∈ wC(Λ, ci′ )Γw−1 . Therefore, by (4), the conjugates of ci′ give rise to
[Ci : wC(Λ, ci′ )Γw−1 ] = [Ci′ : C(Λ, ci′ )Γ] = [C(θ(Λ), θ(ci′ )) : θ(C(Λ, ci′ ))]
empty period cycles in Γσ and so the result follows.
Remark 3.4. The algebraic type of the centralizers of reflections was
found by Singerman in his thesis [15] (see also [16]). However, what
made Theorem 3.3 effective, is the fact that by going a bit more into
details in the Singerman’s papers, one can find explicit generators for
these groups (e.g. [2], [9]).
Fixed Points of Automorphisms of Klein Surfaces
79
4. Some examples
We finish the paper by developing some examples in which we find
topological type of the fixed-points sets for all automorphisms for some
extremal actions of finite groups on non-orientable, unbordered Klein
surfaces. Let ν(g) be the largest possible number of automorphisms of a
non-orientable unbordered Klein surface of topological genus g. Then we
have the following result due to Conder, Maclachlan, Todorovic Vasiljevic and Wilson.
Theorem 4.1 ([6]). If g is odd then ν(g) ≥ 4g. If g is even then
ν(g) ≥ 8(g − 2). Furthermore, these bounds are sharp for infinitely
many g.
Example 4.2. The action of order 4g from Theorem 4.1 is given in [6]
by the epimorphism
θ : Λ → G = D2g = hu, v | u2 , v 2g , (uv)2 i,
where Λ is an NEC-group with signature (0; +; [−]; {(2, 2, 2, g)}) and
θ(c0 ) = u,
θ(c1 ) = uv g ,
θ(c2 ) = v g ,
θ(c3 ) = uv 2 .
Corollary 4.3. The following table gives the topological structures of the
sets of points fixed by automorphisms acting on extremal Klein surfaces
from Example 4.2
Representative
of a conjugacy class
Isolated
fixed points
Ovals
u
2
1
v
uv g
2g
0
1
1
uv g+2
v 2α
2
4
0
0
g
Automorphisms from the remaining conjugacy classes have no fixed
points.
Proof: Here ci , ci+1 ∈ C(Λ, ci ) for i = 1, 2 and so θ(C(Λ, ci )) has at least
4 elements. Also, c1 , c2 , c3 ∈ C(Λ, c2 ) and so θ(C(Λ, c2 )) = D2g . So
using our formulas and some obvious facts concerning normalizers and
centralizers in dihedral groups, we obtain the above topological structure
of the set of fixed points of all elements of G.
80
G. Gromadzki
Example 4.4. The case of even g is technically more involved. Here,
the action of order 8(g − 2) was given in [6] in two steps. First, let
Λ be an NEC-group with signature (0; +; [−]; {(2, 2, 2, 4)}) and consider
homomorphism
θ : Λ → H = Zg−2 ⋊ (D4 × Z2 ) = hwi ⋊ hu, v | u2 , v 4 , (uv)2 i × hti ,
given by
θ(c0 ) = wu,
θ(c1 ) = v 2 ,
θ(c2 ) = t,
θ(c3 ) = uv,
where each of the generators u, v, t conjugates w to its inverse. Then,
the image G of θ has order 8(g − 2) and it acts on a non-orientable
unbordered Klein surface of topological genus g.
Corollary 4.5. The following table gives the topological structures of the
sets of points fixed by automorphisms acting on extremal Klein surfaces
from Example 4.4
Representative
of a conjugacy class
Isolated
fixed points
Ovals
wu
v2
0
2(g − 2)
1
2
t
uv
0
0
2
g/2 − 1
wuv 2
v2 t
tuv
vw
(
4
8
4
8
g ≡ 0 (4)
g 6≡ 0 (4)
4
0
0
0
0
Automorphisms from the remaining conjugacy classes have no fixed
points.
Proof: Now,
C(Λ, c0 ) = hc0 i ⊕ hc1 i ∗ h(c0 c3 )2 i,
C(Λ, c3 ) = hc3 i ⊕ hc2 i ∗ h(c0 c3 )2 i,
C(Λ, ci ) = hci i ⊕ (hci−1 i ∗ hci+1 i) for i = 1, 2
Fixed Points of Automorphisms of Klein Surfaces
81
by [16] (see also [2], [9]). Furthermore the images of c0 , c1 , c2 , c3 ,
c1 c2 , c2 c3 , c0 c3 are pairwise non-conjugate in G and straightforward, but
rather tedious, calculations of their centralizers give the above
results.
Acknowledgements. The author is deeply grateful to the referee for
her/his inquiring questions, very accurate and helpful comments which
allowed to improve initially submitted version of the paper a lot.
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Institute of Mathematics
University of Gdańsk
Wita Stwosza 57
80-952 Gdańsk
Poland
E-mail address: [email protected]
Primera versió rebuda el 30 de juliol de 2007,
darrera versió rebuda el 14 d’abril de 2008.